The case \(\beta =0\)  of Theorem REF ,
but with respect to Whitney cube decompositions,
was established in [1]}.
The constant \(1/53\)  in the statement of Theorem REF 
is not that essential and it has not been optimized. In fact,
it could be replaced by a larger number (up to \(1/3\) ) by using
a further covering argument.
